The most popular version of partial compositeness is called the anarchic approach to flavor, where the underlying Yukawa couplings of the composites Yu,d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{u,d}$$\end{document} are generic O(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{O}(1)$$\end{document} numbers without any structure. The flavor hierarchy in this case arises due to the hierarchical nature of the mixings between the elementary and the composite states fL,R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\mathrm{L,R}$$\end{document}, due to large anomalous dimensions of the composite operators OL,R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{O}_\mathrm{L,R}$$\end{document}. In this case the mixing is expected to be given by4.5fL,R(ΛC)∼fL,R(ΛF)ΛCΛFdL,R-5/2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_\mathrm{L,R}(\Lambda _\mathrm{C}) \sim f_\mathrm{L,R}(\Lambda _\mathrm{F}) \left( \frac{\Lambda _\mathrm{C}}{\Lambda _\mathrm{F}}\right) ^{d_\mathrm{L,R}-5/2}, \end{aligned}$$\end{document}where dL,R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_\mathrm{L,R}$$\end{document} are the scaling dimensions of the composite operators, and fL,R(ΛF)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\mathrm{L,R}(\Lambda _\mathrm{F})$$\end{document} are the values of the mixing parameters at the flavor scale ΛF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _\mathrm{F}$$\end{document}. A hierarchical flavor structure arises naturally for O(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{O}(1)$$\end{document} anomalous dimensions. The CKM mixing matrix arises from the diagonalization of the anarchic Yukawa matrices (4.4) resulting in hierarchic left and right rotation matrices for the up and down sectors Luij∼Ldij∼min(fqi/fqj,fqj/fqi),Ru,dij∼min(fu,di/fu,dj,fu,dj/fu,di)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_u^{ij} \sim L_d^{ij} \sim \mathrm{min} (f_q^i/f_q^j, f_q^j/f_q^i), R_{u,d}^{ij} \sim \mathrm{min} (f_{u,d}^i/f_{u,d}^j, f_{u,d}^j/f_{u,d}^i)$$\end{document}. This results in a hierarchical CKM matrix completely determined by the mixing of the LH states, and with the relations fq1/fq2∼λ,fq2/fq3∼λ2,fq1/fq3∼λ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_q^1/f_q^2 \sim \lambda , f_q^2/f_q^3 \sim \lambda ^2, f_q^1/f_q^3 \sim \lambda ^3$$\end{document} (where λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is the Cabibbo angle), while the diagonal quark masses are given by mu,di=fqifu,div\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{u,d}^i = f_q^i f_{u,d}^i v$$\end{document}.

One of the consequences of this mechanism is that for states where the mixing is close to maximal, the mass of the heavy state must be well below the compositeness scale ΛC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _\mathrm{C}$$\end{document}. We can understand this by considering the interplay between a single composite fermion multiplet with mass mΨ=gΨf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\Psi } = g_{\Psi } f$$\end{document} and its couplings λL,R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _\mathrm{L,R}$$\end{document} with the elementary fermions ψL,R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _\mathrm{L,R}$$\end{document}. The mixing parameter is given by4.6fL,R=λL,RλL,R2+gΨ2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_\mathrm{L,R} = \frac{\lambda _\mathrm{L,R}}{\sqrt{\lambda _\mathrm{L,R}^2 + g_{\Psi }^2}} \ . \end{aligned}$$\end{document}For this to approach unity we need gΨ≪4π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\Psi \ll 4\pi $$\end{document}, in agreement with our original expectation that the state responsible for cutting off the quadratic dependence of the Higgs potential should appear well below the cutoff scale.

Flavor violations in this anarchic scenario are protected by the RS-GIM mechanism [87], which is simply the fact that every flavor violation must go through the composite sector; thus, all flavor violating operators will be suppressed by the appropriate mixing factors. For example, a typical ΔF=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta F=2$$\end{document} 4-Fermi operator mediated by a composite resonance of mass mρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\rho $$\end{document} and coupling gρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\rho $$\end{document}, will have the structure4.7fqifq†jfqkfql†gρ2mρ2q¯iqjq¯kql,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_q^i f_q^{\dagger j} f_q^k f_q^{l \dagger } \frac{g_\rho ^2}{m_\rho ^2} \bar{q}^i q^j \bar{q}^k q^l , \end{aligned}$$\end{document}leading to a quark-mass dependent suppression of these operators. As we will review in Sect. 6.2, the RS-GIM mechanism with completely anarchic Yukawa couplings is not sufficient to avoid the stringent flavor constraints from the kaon system or from several dipole operators, pushing the compositeness scale f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f$$\end{document} to the multi-TeV regime.

Another possible way of protecting the flavor sector from large corrections is by imposing a flavor symmetry on the composite sector. In this case we will lose the explanation of the origin of the flavor hierarchy; however, we might be able to obtain a setup that is minimally flavor violating (MFV), or next-to-minimally flavor violating (NMFV). This was first carried out in the extra dimensional context in [88–90], and later it was implemented in the four dimensional language in [91, 92]. The flavor symmetry structure is determined by the flavor structure of the mixing matrices λL,R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _\mathrm{L,R}$$\end{document} as well as the composite Yukawa matrices Yu,d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{u,d}$$\end{document}. A flavor invariance of the composite sector will imply that the composite Yukawas are proportional to the unit matrix Yu,d∝\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{u,d} \propto $$\end{document} Id3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_3$$\end{document} for the case with maximal U(3)3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{U }(3)^3$$\end{document} flavor symmetry in the composite sector. In order to have MFV, we need to make sure that the only sources of flavor violation are proportional to the SM Yukawa couplings. The simplest possibility is to make the LH mixing matrix proportional to the unit matrix, and the RH mixing matrices proportional to the up- and down-type SM Yukawa couplings:4.8λL∝Id3,λRu∝yu,λRd∝yd.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _\mathrm{L} \propto \mathrm{Id}_3, \ \ \lambda _{Ru} \propto y_u, \ \ \lambda _{Rd} \propto y_d. \end{aligned}$$\end{document}This scenario corresponds to the case with composite left-handed quarks and elementary right-handed quarks, and an explicit implementation of MFV. However, the fact that the left-handed quarks are composite will imply potentially large corrections to electroweak precision observables. The other possibility is to introduce the flavor structure in the left-handed mixing matrix. In order to be able to reproduce the full CKM structure, one needs to double the partners of the LH quarks to include Qu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_u$$\end{document} and Qd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_d$$\end{document}: the composite Yukawa of Qu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_u$$\end{document} will give rise to up-type SM Yukawa couplings, while those of Qd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_d$$\end{document} to down-type Yukawas, while their mixings λLu,λLd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{Lu},\lambda _{Ld}$$\end{document} are proportional to the SM Yukawas. Hence the ansatz for right-handed compositeness is4.9λLu∝yu,λLd∝yd,λRu∝Id3,λRd∝Id3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _{Lu} \propto y_u, \ \ \lambda _{Ld} \propto y_d, \ \ \lambda _{Ru} \propto \mathrm{Id}_3, \ \ \lambda _{Rd} \propto \mathrm{Id}_3, \end{aligned}$$\end{document}which is also an implementation of MFV.

In the MFV scenarios discussed above the composite sector has a U(3)3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{U }(3)^3$$\end{document} flavor symmetry, and either the LH or RH quarks are substantially composite, the degree fixed such as to reproduce the Yukawa coupling of the top. However, the light quarks appear to be very SM-like, more so after LHC dijet production measurements pp→jj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$pp \rightarrow jj$$\end{document} in agreement with the SM, and it might be advantageous to reduce the flavor symmetry, allowing only the third generation quarks to be composites. Furthermore, the models with large flavor symmetries can significantly influence the predictions for the Higgs potential. If parts of the first and second generation are largely composite, along with that of the third, their contributions to the Higgs potential will be enhanced beyond the usual expectations. Accordingly, the phenomenology of the fully MFV models can be significantly modified, as we comment in Sect. 6. A lot of effort has been put recently into exploring the models where the third generation is split from the first two. This next-to-minimal flavor violation corresponds to imposing a U(2)3×U(1)3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{U }(2)^3 \times \mathrm{U }(1)^3$$\end{document} or U(3)2×U(2)×U(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{U }(3)^2 \times \mathrm{U }(2) \times \mathrm{U }(1)$$\end{document} flavor symmetry on the composite sector: it is phenomenologically viable or even favored [92–94], keeping the natural expectations that the Higgs potential is saturated by the top and its partners. We will discuss the main phenomenological signatures of these scenarios in Sect. 6.2.

Finally, there are other possibilities to reproduce the flavor structure of the SM while avoiding the constraints from flavor observables. These rely as well on flavor symmetries. One scenario, originally proposed in [88], is to assume that all the mixing matrices λL,R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _\mathrm{L,R}$$\end{document} are proportional to the identity, while all the flavor structure is provided by the composite sector, that is, Yu,d∝yu,d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{u,d} \propto y_{u,d}$$\end{document}. This setup satisfies the rules of MFV, and all the SM quarks must have a large degree of compositeness.

One last logical possibility to comply with experiments is that the composite sector respects CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP$$\end{document}, given that most of the bounds come from CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP$$\end{document}-violating observables. In this case the Yukawa couplings of the composite sector can be chosen to be real matrices, while the mixings introduce non-negligible CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP$$\end{document} phases if the SM fermions are coupled to more than one composite operator. It has been shown in [91] that this idea might give rise to a realistic theory of flavor.

Most of the new physics effects are of the ‘universal type’ and can be encoded in the modifications of the SM gauge bosons’ two-point functions [119–121]. The most relevant effects in each class can be parametrized by the parameters11S^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{S}$$\end{document}, T^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{T}$$\end{document}, W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W$$\end{document}, and Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y$$\end{document}, where the first two generically yield the most stringent constraints, since the other two are typically suppressed by extra powers of g2/g∗2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g^2/g^2_*$$\end{document}.

There are two generic contributions to the S^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{S}$$\end{document} parameter which arise in all composite Higgs models: the UV contribution from heavy spin-1 resonances that can be estimated as6.1S^UV∼mW2mρ2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{S}_\mathrm{UV}\sim \frac{m_W^2}{m_\rho ^2}\,, \end{aligned}$$\end{document}and an IR contribution associated with the reduced Higgs coupling cV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_V$$\end{document} to the EW gauge bosons [42]. This second one can be understood as follows. For mh≫mZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_h\gg m_Z$$\end{document}, the S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S$$\end{document}-parameter in the SM scales logarithmically with the Higgs mass as result of a cancelation of the log-divergent one-loop contributions of virtual Goldstone and Higgs bosons, logmh/mZ=logΛ/mZ-logΛ/mh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log m_h/m_Z=\log \Lambda /m_Z -\log \Lambda /m_h$$\end{document}. In composite Higgs models, while the Goldstone boson loop stays the same as in the SM, the Higgs boson loop is reduced and hence the cancelation is spoiled, leaving over logΛ/mZ-cV2logΛ/mh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log \Lambda /m_Z -c_V^2 \log \Lambda /m_h$$\end{document}. Thus the S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S$$\end{document}-parameter becomes logarithmically sensitive to the new physics scale Λ∼mρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \sim m_\rho $$\end{document} to be identified with the masses of the heavy resonances (of spin 0, 1, or 2) that couple to the W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W$$\end{document} and the Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z$$\end{document} [42]6.2S^IR≃S^SM(mheff)=g296π2logmheffmZ,mheff=mhΛmh1-cV2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{S}_\mathrm{IR}&\simeq \hat{S}_\mathrm{SM}(m^{\text {eff}}_h)=\frac{g^2}{96\pi ^2}\log \left( \frac{m_{h}^{\text {eff}}}{m_Z}\right) ,\nonumber \\&m_{h}^{\text {eff}}=m_h\left( \frac{\Lambda }{m_h}\right) ^{1-c_V^2}. \end{aligned}$$\end{document}Using a dispersion relation approach [122] one can refine these estimates and achieve a O(mh/mρ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(m_h/m_\rho )$$\end{document} accuracy in S^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{S}$$\end{document} at leading order in g2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g^2$$\end{document} if the spectral density of the strong sector is known. For example, using vector meson dominance as in [115, 122], one finds6.3S^=g296π2v2f2logmρ125GeV-0.29+mW2f2fρ2mρ2-fa2ma2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{S}\!=\!\frac{g^2}{96\pi ^2}\frac{v^2}{f^2}\left( \log \frac{m_\rho }{125\text {GeV}}\!-\!0.29\right) \!+\! \frac{m_W^2}{f^2}\left( \frac{f_\rho ^2}{m_\rho ^2}-\frac{f_a^2}{m_a^2}\right) ,\nonumber \\ \end{aligned}$$\end{document}where fρ,a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{\rho ,a}$$\end{document} and mρ,a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\rho ,a}$$\end{document} denote the decay constants and the masses of vector and axial resonances. The new physics contribution to S^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{S}$$\end{document} can be kept under control if mρ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\rho ^2$$\end{document} is sufficiently large, although this generically introduces some tuning in the Higgs potential, since mρ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\rho ^2$$\end{document} fixes the scale where gauge-loop contributions are cut off. Another option is to invoke some degree of cancelation between different contributions directly in S^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{S}$$\end{document}, for instance coming from extra scalars or fermions [123], although these are loop suppressed and generically model dependent.12 Moreover, in [113] it was pointed out that fermion loops in composite Higgs models may provide additional sources of logarithmically enhanced contributions that can be understood in terms of the running of the two dimension-6 operators OW,B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}_{W, B}$$\end{document} related to S^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{S}$$\end{document} [41].

It was recognized long ago [126] that the T^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{T}$$\end{document}-parameter can be protected against new physics contributions by a custodial symmetry SU(2)C⊂SO(4)≅SU(2)L×SU(2)R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SU }(2)_\mathrm{C} \subset \mathrm{SO }(4) \cong \mathrm{SU }(2)_\mathrm{L} \times \mathrm{SU }(2)_\mathrm{R}$$\end{document}. This requires that the new sector respects custodial symmetry to a very high degree, most often forbidding new sources of breaking beyond those already present in the SM, that is, the Yukawa coupling of the top and the hypercharge gauge coupling. In particular, it is required that the new states cutting off the Higgs potential, in particular the vector partners, come in complete representations of SO(4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SO }(4)$$\end{document}. This has been explicitly verified in many little Higgs models, see for instance [127–129]. In holographic Higgs models this requirement is satisfied by construction, since the partners always come in complete representations of the unbroken global symmetry subgroup, which contains SO(4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SO }(4)$$\end{document} [130, 131]. In addition, while the custodial SO(4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SO }(4)$$\end{document} is sufficient to protect the T^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{T}$$\end{document}-parameter when a single Higgs field breaks the electroweak symmetry spontaneously, as we discussed in Sect. 5 this is not the case when extra scalar fields charged under SO(4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SO }(4)$$\end{document} are present, additional Higgs doublets, triplets, etc. In these cases, an ‘enlarged’ custodial symmetry is required (see [96] for a detailed explanation of the THDM case).

With custodial protection, the leading corrections to T^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{T}$$\end{document} arise thus at one loop. Analogously to the case for S^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{S}$$\end{document}, there is a universal IR contribution from the reduced coupling of the Higgs boson which can again be estimated in the heavy Higgs limit as6.4T^IR≃-3g′232π2logmhmZΛmh1-cV2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{T}_\mathrm{IR}\simeq -\frac{3g^{\prime \,2}}{32\pi ^2}\log \left[ \frac{m_h}{m_Z}\left( \frac{\Lambda }{m_h}\right) ^{1-c_V^2}\right] \ . \end{aligned}$$\end{document}These IR contributions due to the modified Higgs couplings, Eqs. (6.2) and (6.4), form a line in the S^-T^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{S}-\hat{T}$$\end{document} plane. If these were the only corrections, then they would imply ξ=v2/f2≲0.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi =v^2/f^2\lesssim 0.1$$\end{document}, see Fig. 2 reproduced from [132].Fig. 2

Confidence-level contours (at 65\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$65$$\end{document}, 95\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$95$$\end{document} and 99%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$99~\%$$\end{document}) for S^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{S}$$\end{document} and T^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{T}$$\end{document} from [132]. The IR contributions alone would imply ξ=v2/f2≲0.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi =v^2/f^2\lesssim 0.1$$\end{document}

Besides the oblique parameters, strongly interacting models usually induce non-universal modifications to the couplings of the top, and due to SU(2)L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SU }(2)_\mathrm{L}$$\end{document} invariance, also to those of the left-handed bottom [134, 135]. This is due to the necessarily large coupling of the top quark to the strong sector, in order to reproduce its large Yukawa coupling. The strongest constraints come from measurements of the ZbLb¯L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z b_\mathrm{L} \bar{b}_\mathrm{L}$$\end{document} coupling, sensitive to the masses of the new-physics states. However, it was shown in [136] that the ZbLb¯L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z b_\mathrm{L} \bar{b}_\mathrm{L}$$\end{document} vertex can be protected from large corrections by a PLR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\mathrm{LR}$$\end{document} parity symmetry, as long as the bL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_\mathrm{L}$$\end{document} embedding does not break it, that is, if bL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_\mathrm{L}$$\end{document} has -1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1/2$$\end{document} charge under both SU(2)L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SU }(2)_\mathrm{L}$$\end{document} and SU(2)R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SU }(2)_\mathrm{R}$$\end{document}.13 As for the custodial symmetry, when this custodial parity is preserved by the strong sector, corrections to ZbLb¯L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z b_\mathrm{L} \bar{b}_\mathrm{L}$$\end{document} can be kept under control. Both symmetries yield important consequences for the quantum numbers and spectrum of the top partner resonances (for instance extended representations such as the 4=(2,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {4} = (\mathbf {2},\mathbf {2})$$\end{document}).

Figure 3 reproduced from [137] shows the best fit region with a small positive δgRb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta g_{Rb}$$\end{document} where the following parametrization is used:146.6L=gcWZμb¯γμgLbSM+δgLBPL+gRbSM+δgRBPRb.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {L}&= \frac{g}{c_W} Z_\mu \bar{b}\gamma ^\mu \left[ \left( g^\mathrm{SM}_{Lb}+\delta g_{LB}\right) P_\mathrm{L}\right. \nonumber \\&+\left. \left( g^\mathrm{SM}_{Rb}+\delta g_{RB}\right) P_\mathrm{R}\right] b. \end{aligned}$$\end{document}The contribution from fermion loops to δgLb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta g_{Lb}$$\end{document} is generically logarithmically divergent as a result of insertions of the mixings that break the PLR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\mathrm{LR}$$\end{document} parity6.7δgLbgLbSM∼yt216π2v2f2logΛ2mΨ2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\delta g_{Lb}}{g^\mathrm{SM}_{Lb}}\sim \frac{y_t^2}{16\pi ^2}\frac{v^2}{f^2}\log \frac{\Lambda ^2}{m_{\Psi }^2}\,. \end{aligned}$$\end{document}Another sensitive test concerns the anomalous coupling of the right-handed top and bottom to the Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z$$\end{document}. This coupling is tightly constrained by b→sγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b \rightarrow s \gamma $$\end{document} measurements. However, the size of the anomalous coupling is generically suppressed by yb/yt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_b/y_t$$\end{document}, yielding mild bounds on the new physics scale, see for instance [138]. Other top related measurements still lack of precision [133, 139].

In the previous sections we have argued that due to its contribution to the Higgs potential, fermionic top partners should be the lightest new physics states. The effects of top-partners on precision tests, which we have reviewed in this section, have been thoroughly discussed in the literature, either in the context of little Higgs models [140], holographic Higgs models [42, 92, 125, 133, 141, 142], or in more generality [132, 143].

Finally, let us again note that modified Higgs couplings to electroweak gauge bosons can be indirectly probed through electroweak precision measurements, Eqs. (6.2) and (6.4). Such modified couplings arise whenever the operator (∂μ(H†H))2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\partial _\mu (H^\dagger H))^2$$\end{document} is generated, to which new physics contributes even if the states responsible for taming the Higgs potential only couple to the Higgs (even if they do not carry electroweak charges in particular). Besides, this operator generically encodes the non-linear self-interactions of the Higgs, intrinsic of its GB nature. As such, it will be suppressed by α/f2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha /f^{2}$$\end{document}, with α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} a numerical factor that depends on the coset structure.

Also note that the case of a dilatonic Higgs needs to be considered separately for the EWPTs. Since a composite Higgs-like dilaton is not embedded into a SU(2) doublet, the argument before does not directly apply. Actually, the couplings of the dilaton to the gauge fields agree with those of the SM Higgs, except for a v/f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v/f$$\end{document} suppression. Thus the corrections to S^IR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{S}_\mathrm{IR}$$\end{document} and T^IR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{T}_\mathrm{IR}$$\end{document} are minimized in the limit v/f→1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v/f \rightarrow 1$$\end{document}, the opposite limit than in ordinary composite Higgs scenarios.

For a recent model independent analysis of the constraints from EWPT, see [144].

Another important direction for taming electroweak precision constraints has been the introduction of T-parity [100, 101]: a Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document} discrete symmetry under which all BSM states are odd. Such a symmetry ensures that all corrections to electroweak precision observables from the new states are at least one-loop suppressed, thus reducing the bounds on the masses of the new states. In this case one can obtain a theory consistent with the electroweak precision observables, even with new states as light as ∼\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim $$\end{document}1TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,\mathrm {TeV}$$\end{document}. T-parity has been one of the leading themes for little Higgs models, and it can of course also be implemented in the general 4D versions.15 An illustration of the electroweak precision observables in a little Higgs model with T-parity can be found in [146].

It has been long known that a simple mechanism to generate the interactions in Eq. (4.1) gives rise also to unsuppressed SM flavor violating 4-Fermi interactions6.8cijklΛF2qiqjq¯kq¯l,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{c^{ijkl}}{\Lambda _\mathrm{F}^2} {q}_i q_j \bar{q}_k \bar{q}_l , \end{aligned}$$\end{document}which generically violate the stringent flavor constraints: for instance from the kaon system, ΛF>103-5TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _\mathrm{F}>10^{3-5} \,\mathrm {TeV}$$\end{document}, while allowing for a sufficiently large top mass one would need ΛF=O(10)TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _\mathrm{F} = \mathcal{O}(10) \,\mathrm {TeV}$$\end{document}. As explained in Sect. 4.1, this tension can be relaxed if the dimension of the operator O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{O}$$\end{document} in Eq. (4.1) is sufficiently close to one, as long as the dimension of O2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{O}^2$$\end{document} does not decrease below four hence reintroducing the hierarchy problem.

It is worth mentioning that other alternatives might be viable, which rely on the flavor dynamics inducing additional suppression of the operators in Eq. (6.8), either via the Yukawa couplings, cijkl∼yu,dijyu,dkl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c^{ijkl} \sim y^{ij}_{u,d} \, y^{kl}_{u,d}$$\end{document}, in which case the bounds on ΛF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _\mathrm{F}$$\end{document} can be relaxed close to the scale required to reproduce the top mass, or effectively imposing MFV, which could be realized if the couplings of the standard model fermions to the strong dynamics arise from the exchange of (supersymmetric) heavy scalars, such as in bosonic technicolor [147–149]. In the former case new physics is to be expected in flavor transitions, while in the latter supersymmetric states remnant of the flavor generation should be observable.

As discussed in Sect. 4.2, the RS-GIM mechanism of partial compositeness significantly reduces the contributions to dangerous flavor transitions. However, it has been shown that the suppression is not quite enough as to provide a fully realistic theory of flavor. Even though ΔF=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta F = 2$$\end{document} 4-Fermi operators6.9fqifq†jfqkfql†gρ2mρ2q¯iqjq¯kql\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_q^i f_q^{\dagger j} f_q^k f_q^{l \dagger } \frac{g_\rho ^2}{m_\rho ^2} \bar{q}^i q^j \bar{q}^k q^l \end{aligned}$$\end{document}are effectively suppressed by four powers of the fermion masses m/v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m/v$$\end{document} or CKM entries VCKM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_\mathrm{CKM}$$\end{document}, measurements of CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP$$\end{document} violation in the kaon system, ϵK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _\mathrm{K}$$\end{document}, put stringent bounds on the LR operators in Eq. (6.9), of the form mρ≳10gρYd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\rho \gtrsim 10 \frac{g_\rho }{Y_d}$$\end{document} TeV [87, 91, 111, 150–152], as well on LL operators. Although less significant, qualitatively similar bounds on LL operators arise from CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP$$\end{document} violation in the B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B$$\end{document} system, mρ≳1gρYuTeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\rho \gtrsim 1 \frac{g_\rho }{Y_u} \,\mathrm {TeV}$$\end{document}. Given the expectation mρ∼gρf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\rho \sim g_\rho f$$\end{document}, these type of constraints bound the combination Yd,uf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{d,u} f$$\end{document}. In explicit constructions of the pGB Higgs, the composite Yukawas Yu,d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{u,d}$$\end{document} are correlated with the masses of the composite fermions cutting off the Higgs potential. These kind of bounds therefore have a significant impact on the fine-tuning. In addition, these bounds have to be contrasted with other potentially problematic flavor observables such as dipole operators6.10fqifq†j116π2Yu,d3mΨ2q¯iσμνFμνqj,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_q^i f_q^{\dagger j} \frac{1}{16 \pi ^2} \frac{Y^3_{u,d}}{m_\Psi ^2} \bar{q}_{i} \sigma _{\mu \nu } F^{\mu \nu } q_{j}, \end{aligned}$$\end{document}generated by loops of composite fermions of mass mΨ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\Psi $$\end{document} and the Higgs. These induce large contributions to b→sγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\rightarrow s\gamma $$\end{document}, direct CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP$$\end{document} violation in ϵ′/ϵK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon '/\epsilon _\mathrm{K}$$\end{document}, and contributions to the flavor conserving electric dipole moment of the neutron, all of them scaling with positive powers of Yd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_d$$\end{document}; thus, the constraints mΨ>αYdTeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\Psi > \alpha Y_d \,\mathrm {TeV}$$\end{document}, with α∼0.5-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \sim 0.5 - 2$$\end{document} [91, 92, 152–154]. All these flavor bounds taken together force the scale of compositeness to be above 2TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$~2 \,\mathrm {TeV}$$\end{document} along with composite couplings g∗=ρ,Ψ≫gSM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{*=\rho ,\Psi } \gg g_\mathrm{SM}$$\end{document}.16

Moreover, let us notice that the operators in Eq. (6.9) could also be mediated by the Higgs or other pGBs (of mass mH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_H$$\end{document}), with the associated enhancement of their coefficients by (mρ2/mh2)(v/f)4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m_\rho ^2/m_h^2)(v/f)^4$$\end{document} or (mρ2/mH2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m_\rho ^2/m_H^2)$$\end{document}, respectively.17 However, it was pointed out in [96, 156] that these unwanted effects can be avoided thanks to the Goldstone nature of these scalars, as long as the embedding of the SM fermions into the global symmetries of the strong sector only allows for a single Yukawa-type operator q¯LqRF(h,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{q}_\mathrm{L} q_\mathrm{R} F(h,H)$$\end{document}, thus enforcing the MFV structure in the scalar interactions.18

This is, however, not the case for a dilatonic Higgs, since the lack of a direct connection with the electroweak VEV generically implies that the fermion mass matrices and the dilaton couplings are misaligned. In that case the best alternative is to assume that the composite sector is endowed with flavor symmetries.

Let us briefly comment on the lepton sector. First of all, given that neutrinos are much lighter than charged leptons, and that their mixings are not hierarchical, it is certainly plausible that neutrino masses come from a different source, or enjoy a different generation mechanism. Factoring out the discussion of neutrino mass generation, the constraints on partial compositeness for leptons with anarchic Yukawas come from [152, 159, 160] the electron EDM, and μ→eγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \rightarrow e \gamma $$\end{document} transitions from penguin mediated dipole operators. The bounds from experimental data are even more stringent than in the quark sector, which makes the minimal implementation of leptonic partial compositeness not viable.

The most appealing option thus seems to rely on lepton flavor global symmetries, enforcing LMFV [161]. Another option to remove tree-level constraints on lepton partial compositeness is by imposing an A4 symmetry on the composite sector [162], alleviating the tension with the loop-induced processes. In that case the degree of compositeness of the leptons must increase in order to yield the proper Yukawa couplings, with the consequence of light tau partners [163].

As review in Sect. 4.2, the scenarios falling into this category can be classified as LH or RH quark compositeness. The degree of compositeness in each case is fixed by the requirement fL,R≳yt/Yu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\mathrm{L,R} \gtrsim y_t/Y_u$$\end{document}, in order to reproduce the top mass. Therefore, in every case the inevitable signal will come from flavor diagonal 4-quark operators,6.11gρ2mρ2fL,R4(q¯γμq)(q¯′γμq′),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{g_\rho ^2}{m_\rho ^2} f_\mathrm{L,R}^4 (\bar{q} \gamma _\mu q) (\bar{q}' \gamma ^\mu q'), \end{aligned}$$\end{document}generated from the exchange of heavy resonances of mass mρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\rho $$\end{document} and coupling gρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\rho $$\end{document}. These have been recently probed at the LHC in pp→jj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$pp \rightarrow jj$$\end{document} angular distributions. The individual bounds for the complete set of independent 4-quark operators, their coefficient normalized to Λ-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda ^{-2}$$\end{document}, range between Λ≳1-5TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \gtrsim 1 -5 \,\mathrm {TeV}$$\end{document} [164]. These place strong constraints on the degree of compositeness of the quarks, given the identification Λ∼f/fL,R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \sim f/f_\mathrm{L,R}^2$$\end{document}, for mρ∼gρf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\rho \sim g_\rho f$$\end{document}. Taking the most favorable situation, that is, fL,R∼yt/Yu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\mathrm{L,R} \sim y_t/Y_u$$\end{document}, the dijets constraints bound the combination Yu2f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_u^2 f$$\end{document}, again implying large partners masses as in the anarchic case.

There is another class of constraints that apply only to LH or RH compositeness. If the LH quarks are composite, their (flavor diagonal) couplings to W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W$$\end{document} and Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z$$\end{document} receive significant corrections, which affect precision observables such as quark–lepton universality in kaon and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} decays or the hadronic width of the Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z$$\end{document} [91].19 The corresponding bounds take the form mΨ≳35fLYuv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\Psi \gtrsim 35 f_\mathrm{L} Y_u v$$\end{document}, which again, taking fL,R∼yt/Yu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\mathrm{L,R} \sim y_t/Y_u$$\end{document}, implies a strong bound on the partners masses mΨ≳35mt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\Psi \gtrsim 35 m_t$$\end{document}. For the case of RH composite quarks, given that their coupling to W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W$$\end{document} and Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z$$\end{document} are still poorly measured (and can be easily protected by their proper embedding into the global symmetries of the strong sector), the previous measurements do not yield important constraints. However, flavor violating LL 4-Fermi operators Eq. (6.9) are still generated with a significant coefficient (yuyu†)2/(f2Yu4fR4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$~ (y_u y_u^\dagger )^2/(f^2 Y_u^4 f_\mathrm{R}^4)$$\end{document} [92], which even though MFV suppressed, still yields Yu2fR2f≳6TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_u^2 f_\mathrm{R}^2 f \gtrsim 6 \,\mathrm {TeV}$$\end{document}. Notice in particular that while this constraint prefers fR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\mathrm{R}$$\end{document} large, the dijet bounds push towards fR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\mathrm{R}$$\end{document} small.

In summary, flavor models with U(3)3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{U }(3)^3$$\end{document} symmetry are under a significant stress from recent measurements of dijet production at the LHC. With the increase of energy at the next run of the LHC, such measurements will provide conclusive results about this possibility.

In models where the flavor symmetry is reduced in order to uncouple the fraction of compositeness of the light generations and that of the top quark, the compositeness constraints from measurements of W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W$$\end{document} and Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z$$\end{document} couplings or dijet production (discussed above), are irrelevant. Therefore in these scenarios the only phenomenologically relevant flavor constraints are the consequences of the third generation (LH chirality, RH, or both) being distinct from the first two. In this case it is important to point out that the R rotation matrices are very close to the identity in all the scenarios, with the corresponding suppression of the most dangerous LR 4-Fermi operators in Eq. (6.9) [92, 94]. Still the most sensitive flavor observables come from the kaon and B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B$$\end{document} systems (and the D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D$$\end{document} system in the case of RH compositeness), as in the anarchic case, but with correlations among them, depending on the particular symmetry implementation. Most importantly, the associated bounds can now be satisfied for relatively low values of f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f$$\end{document} or the partner masses. This makes the U(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{U }(2)$$\end{document} scenarios the most favored ones for a natural electroweak scale, while still offering good prospects of new physics effects in flavor physics.

Let us conclude this section by commenting on the particulars of little Higgs models. Although their UV completion is not a priori determined, thus making an assessment of flavor and CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP$$\end{document} violation more model dependent, solely from the interactions of the low energy degrees of freedom valuable lessons can be inferred, which are of course similar to those discussed in this section. Gauge and top partners contribute to neutral meson mixing and CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CP$$\end{document} violation, with bounds at the same level or in some cases milder than those coming from EWPT [165–171] (and see [140] for a recent review on the top partners effects).

After the Higgs discovery, one of the major enterprises in particle physics has been the extraction of the linear couplings of the Higgs to the other SM fields. These are obtained by fitting the experimental data on σ×BR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \times BR$$\end{document}, see [172–174] and references therein. The best tested Higgs couplings to date are those to electroweak gauge bosons hZZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$hZZ$$\end{document} and hWW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$hWW$$\end{document} (with less precision), and to massless gauge bosons hgg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$hgg$$\end{document} and hγγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\gamma \gamma $$\end{document}, induced at one loop in the SM. Indirectly, through its contribution to hgg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$hgg$$\end{document} and hγγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\gamma \gamma $$\end{document}, the coupling to top quarks, htt¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h t \bar{t}$$\end{document} is also being tested. The first results on the coupling to tau leptons hττ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h \tau \bar{\tau }$$\end{document} and bottom quarks hbb¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h b \bar{b}$$\end{document} have also been obtained.

In order to make connection with the experimental data and compare with different models, we parametrize the linear interactions of the Higgs by the following Lagrangian:6.12Leff(h)=cV2mW2Wμ+W-μ+mZ2Zμ2-ctmtt¯t-cbmbb¯b-cτmττ¯τhv+cγγ2AμνAμν+cZγZμνγμν+cgg2GμνaGa,μνhv,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {L}_{eff}^{(h)}&= \left( c_V \left( 2 m_W^2 W^{+}_\mu W^{-\mu } + m_Z^2 Z_\mu ^2 \right) \right. \nonumber \\&\left. - c_t m_{t} \bar{t} t- c_b m_{b} \bar{b} b - c_\tau m_{\tau } \bar{\tau } \tau \right) \frac{h}{v}\nonumber \\&+\left( \frac{c_{\gamma \gamma }}{2} A_{\mu \nu } A^{\mu \nu } + c_{Z\gamma } Z_{\mu \nu } \gamma ^{\mu \nu }+ \frac{c_{gg} }{2} G^{a}_{\mu \nu } G^{a,\mu \nu }\right) \frac{h}{v},\nonumber \\ \end{aligned}$$\end{document}and present in Table 2 the predictions for two distinct composite Higgs models, the SO(5)/SO(4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SO }(5)/\mathrm{SO }(4)$$\end{document} model of [31], known as the Minimal Composite Higgs Model (MCHM), and the dilatonic Higgs following [63]. For the MCHM, we only include the predictions associated to the GB non-linear nature of the Higgs, dictated by the symmetry structure of the model, and we comment on the effects of the light SM partners below, which in any case give subleading corrections. For the case of the dilaton the couplings are entirely determined by scale invariance and its breaking.Table 2

Coefficients of the linear Higgs couplings in Eq. (6.12), for the SM, the SO(5)/SO(4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SO }(5)/\mathrm{SO }(4)$$\end{document} composite Higgs (MCHM), and the dilaton Higgs